First, consider the distribution of set sequences \(V(G)=U_0\supseteq \cdots \supseteq U_N\)  obtained as follows: Set \(U_0=V(G)\) . For as long as \(|U_i|>n^{1/(r+1)}\) , for each \(j\in [r]\)  let \(U_{i+1}\cap V_j\)  be a uniformly random subset of \(U_i\cap V_r\)  of size exactly \(\lceil \eta |U_{i+1}|/r\rceil \) . Observe that REF , REF , and REF  hold by definition. Chernoff's inequality for hypergeometric distributions (lem:chernoff) and a union bound imply that REF  holds w.h.p.
Showing that REF  holds w.h.p. requires a little more work. Let \(i \in [N]\)  and \(j,k \in [r]\)  be distinct. We use regularity and Chernoff's inequality to show that with all but exponentially small probability there holds \(d(U_i \cap A_j, U_i \cap A_k) = d_{j,k} \pm 2\varepsilon \) . In order to show that \(G[U_i \cap A_j,U_i \cap A_k]\)  is \(\varepsilon ^{1/20}\) -regular we use the well-known equivalence between the sums of codegrees and regularity [1]}. That is, we first use McDiarmid's inequality (lem:McDiarmid) to prove that with all but exponentially small probability there holds
\(\sum _{u,v \in U_i \cap A_j} |\lbrace w \in U_i \cap A_k : uw,vw \in E(G)\rbrace | = (1\pm o(1)) \left( \frac{|U_i|}{|V|} \right)^3 \sum _{u,v \in A_j} |\lbrace w \in A_k : uw,vw \in E(G)\rbrace |.\) 
Since \((A_j,A_k)\)  is \((d_{j,k},\varepsilon )\) -regular the sum on the right is equal to \(n^3d_{j,k}^2 \pm \varepsilon ^{1/4}n^3\) . Hence the sum on the left is equal to \(|U_i \cap A_j|^3 d_{j,k}^2 \pm \varepsilon ^{1/5}|U_i|^3\) . But this, in turn, implies that \(G[U_i \cap A_j,U_i \cap A_k]\)  is \(\varepsilon ^{1/20}\) -regular. In order to show super-regularity it remains to verify the minimum degree condition; this follows immediately from REF .
Next, let \(\lbrace v_1,\ldots ,v_m\rbrace \subseteq V(G)\)  and \(\vec{x}\in \lbrace 0,\ldots ,N\rbrace ^m\) . Clearly
\(\mathbb {P}\bigg [\bigwedge _{i=1}^m(v_i\in U_{x_i})\bigg ]\le \prod _{i=1}^m \frac{3|U_{x_i}|}{2n},\) 
since each next set \(U_{i+1}\)  of the binomial set process can be coupled inside a \(\eta (1+1/N^2)\) -binomial random subset, say, of the current set \(U_i\)  (and multiplying over at most \(N\)  steps).
Let \(\mathcal {E}\)  be the event that properties itm:super-vortex-balanced,itm:super-vortex-set-sizes,itm:super-vortex-final-set-size,itm:super-vortex-super-regular,itm:super-vortex-high-degrees hold for \(U_0,\ldots ,U_N\) . Let \(V_0\supseteq \cdots \supseteq V_N\)  be the distribution obtained by conditioning \(U_0\supseteq \cdots \supseteq U_N\)  on the occurrence of \(\mathcal {E}\) . By definition, \(V_0\supseteq \cdots \supseteq V_N\)  satisfies properties itm:super-vortex-balanced,itm:super-vortex-set-sizes,itm:super-vortex-final-set-size,itm:super-vortex-super-regular,itm:super-vortex-high-degrees. Furthermore, for every nonempty \(\lbrace v_1,\ldots ,v_m\rbrace \subseteq V(G)\)  and \(\vec{x}\in \lbrace 0,\ldots ,N\rbrace ^m\) :
\(\mathbb {P}\bigg [\bigwedge _{i=1}^m(v_i\in V_{x_i})\bigg ] = \mathbb {P}\bigg [\bigwedge _{i=1}^m(v_i\in U_{x_i})\bigg |\mathcal {E}\bigg ]\le \frac{\mathbb {P}\big [\bigwedge _{i=1}^m(v_i\in U_{x_i})\big ]}{\mathbb {P}[\mathcal {E}]}\le \prod _{i=1}^m\frac{2|V_{x_i}|}{n},\) 
as desired. The last inequality comes from applying itm:super-vortex-set-sizes iteratively at most \(N\)  times.
